Modeling of Conditional Index Changes


An electrical circuit containing switch elements represents a variable structure system. The structure of the circuit is determined by two possible switch positions. The behavior of a switch element can be described by a switch equation using a discrete variable to determine the switch position. However the causality of a switch element cannot be fixed. Commercial simulation programs prevent usually the causality problem for switch elements in a tricky way. A closed switch element is replaced by a high conductance while an opened switch element is replaced by a high resistance. Yet this easy unideal solution goes along with the disadvantage of creating unnatural stiffness in the model. A conditional index system resulting from switching in a physical system requires fixed causality assignments that cause conflicts with ideal switch elements.

The idea to resolve the causality assignment was to modify Pantelides Algorithm to a suitable formulation for conditional index changes by modifying the corresponding switch equations. However this work shows that the modification concept cannot solve an easy example. Hence it can be concluded that merely modifying the switch equations does not bring us closer to the desired goal, the formulation of a single model of an ideal switching circuit involving conditional index changes that can be simulated in all switch positions.

However the work resulted in a new idea, the use of implicit difference formulae that makes the modifications of switch equations unnecessary. The difference formulae widely used in commercial DAE solvers substitute the original derivatives in inductor and capacitor equations. The new concept is used to simulate the earlier mentioned easy example and a more complex circuit for train speed control. The mathematical description for the complex circuit pointed out that there are remaining singular cases. These singular cases are investigated and can be explained by the nature of an ideal switch simulation. In the sequel several possibilities to prevent these singularities are introduced and explained.